双椭圆曲面上半稳定层模空间的拓扑与几何
Topology and geometry of moduli spaces of semistable sheaves on bielliptic surfaces
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中文总结 AI 辅助
研究双椭圆曲面上半稳定层模空间的拓扑与低次霍奇理论，通过特定纤维化等方法构造满同态计算相关贝蒂数，扩展计算到正秩向量，证明\(H^2(F,\mathbb{C})\)性质，在一定条件下得出\(F\)的相关结论及\(M^\circ_{H,S}(\mathbf{v})\)的性质。
中文摘要 AI 辅助
我们研究双椭圆曲面上半稳定层模空间的拓扑和低次霍奇理论。对于满足正性条件\(\operatorname{nt}(L)\geq3\)的本原秩为零的 Mukai 向量\(\mathbf{v}=(0,c_1(L),\chi)\)，特殊的固定行列式分量\(M_{H,S}(\mathbf{v},L)\)允许一个到\(|L|\)的支撑态射，并被解释为相对紧化雅可比。利用这个纤维化、正线性系统的莱夫谢茨型性质、单值化和混合霍奇结构，我们构造了一个满同态\(\pi_1(S)\twoheadrightarrow\pi_1\bigl(M_{H,S}(\mathbf{v},L)\bigr)\)，并计算了 Albanese 纤维\(F\)、\(M_{H,S}(\mathbf{v},L)\)和特殊分量\(M^\circ_{H,S}(\mathbf{v})\)的前两个贝蒂数。傅里叶 - 穆凯变换和布里奇兰德壁穿越将这些计算扩展到正秩的本原可允许 Mukai 向量。我们进一步证明\(H^2(F,\mathbb{C})\)是纯霍奇型\((1,1)\)。如果\(\lambda_S=\ell(\mathbf{v})=1\)，那么\(F\)在有限拟 étale 覆盖下是严格不可约的 Calabi - Yau 簇。在对拉回极化的额外一般性假设下，\(M^\circ_{H,S}(\mathbf{v})\)是光滑的，并且与\(\operatorname{Pic}^0(S)\times\operatorname{Hilb}^{\mathbf{v}^2/2}(S)\)非典范双有理。
英文摘要
We study the topology and low-degree Hodge theory of moduli spaces of semistable sheaves on bielliptic surfaces. For primitive rank-zero Mukai vectors $\mathbf{v}=(0,c_1(L),χ)$ satisfying the positivity condition $\operatorname{nt}(L)\geq3$, the distinguished fixed-determinant component $M_{H,S}(\mathbf{v},L)$ admits a support morphism to $|L|$ and is interpreted as a relative compactified Jacobian. Using this fibration, Lefschetz-type properties of positive linear systems, monodromy, and mixed Hodge structures, we construct a surjective homomorphism $ π_1(S)\twoheadrightarrowπ_1\bigl(M_{H,S}(\mathbf{v},L)\bigr) $ and compute the first two Betti numbers of an Albanese fiber $F$, $M_{H,S}(\mathbf{v},L)$, and the distinguished component $M^\circ_{H,S}(\mathbf{v})$. Fourier-Mukai transforms and Bridgeland wall crossing extend these computations to primitive admissible Mukai vectors of positive rank. We further prove that $H^2(F,\mathbb{C})$ is of pure Hodge type $(1,1)$. If $λ_S=\ell(\mathbf{v})=1,$ then $F$ is a strict irreducible Calabi-Yau variety up to a finite quasi-étale cover. Under the additional genericity assumption on the pullback polarization, $M^\circ_{H,S}(\mathbf{v})$ is smooth and is noncanonically birational to $\operatorname{Pic}^0(S)\times\operatorname{Hilb}^{\mathbf{v}^2/2}(S)$.
